If one real root of the quadratic equation

$81 x^{2}+k x+256=0 ; x=\alpha, \alpha^{3}$

$\Rightarrow \alpha^{4}=\frac{256}{81} \Rightarrow \alpha=\pm \frac{4}{3}$

Now $-\frac{\mathrm{k}}{81}=\alpha+\alpha^{3}=\pm \frac{100}{27}$

$\Rightarrow k=\pm 300$